Benchmark Example No.

17

Stress Calculation at a Rectangular Prestressed

Concrete CS

SOFiSTiK | 2023
 VERiFiCATiON
DCE-EN17 Stress Calculation at a Rectangular Prestressed Concrete CS

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Front Cover
Volkstheater, Munich Photo: Florian Schreiber
Stress Calculation at a Rectangular Prestressed Concrete CS

Overview

Design Code Family(s): DIN

Design Code(s): DIN EN 1992-1-1
Module(s): AQB, TENDON
Input file(s): stress prestress.dat

1 Problem Description
The problem consists of a rectangular cross-section of prestressed concrete, as shown in Fig. 1. The
stresses developed at the section due to prestress and bending are verified.

h d My
Np
zp
Ap

Figure 1: Problem Description

2 Reference Solution
This example is concerned with the design of prestressed concrete cs, subject to bending and prestress
force. The content of this problem is covered by the following parts of DIN EN 1992-1-1:2004 [1]:

• Stress-strain curves for concrete and prestressing steel (Section 3.1.7, 3.3.6)

• Verification by the partial factor method - Design values (Section 2.4.2)

• Prestressing force (Section 5.10.2, 5.10.3)

NP / A NP · zp / W2 M/ W2

+ + =

zp
Ap

NP / A NP · zp / W1 M/ W1

Figure 2: Stress Distribution in Prestress Concrete Cross-section

In rectangular cs, which are prestressed and loaded, stress conditions are developed, as shown in Fig.
2, where the different contributions of the loadings can be seen. The design stress-strain diagrams for
prestressing steel is presented in Fig. 3, as defined in DIN EN 1992-1-1:2004 [1] (Section 3.3.6).

SOFiSTiK 2023 | Benchmark No. 17 3

 Stress Calculation at a Rectangular Prestressed Concrete CS

A
σp
ƒpk

ƒp0,1k
ƒpd = ƒp0,1k / γs

B A Idealised

B Design

εp

Figure 3: Idealised and Design Stress-Strain Diagram for Prestressing Steel

3 Model and Results

The simply supported beam of Fig. 4, consists of a rectangular cross-section with properties as defined
in Table 1 and is prestressed and loaded with its own weight. A verification of the stresses is performed
in the middle of the span with respect to DIN EN 1992-1-1:2004 (German National Annex) [1], [2]. The
geometry of the tendon can be visualised in Fig. 5. The calculation steps [3] are presented below and
the results are given in Table 2.

Table 1: Model Properties

Material Properties Geometric Properties Loading (at  = 10 m)

C 35/ 45 h = 100.0 cm Mg = 1250 kNm

Y 1770 b = 100.0 cm Np = −3651.1 kN

d = 95.0 cm
L = 20.0 m

Ap = 28.5 cm2

Figure 4: Simply Supported Beam

4 Benchmark No. 17 | SOFiSTiK 2023

 0.916
0.973
0.000

1
0.916 0.00
3572.3 0.057
0.917
3574.2
0.917 0.057
1.00
3576.1 0.057
0.918


3578.1


0.918 0.114
2.00



V
3580.2 0.057
0.919
3582.6
0.171

Case
0.920 3.00
3585.6 0.055
0.920

SOFiSTiK 2023 | Benchmark No. 17

3588.9
0.921 0.223
4.00
3592.4 0.050
0.922
3596.1
0.923 0.271
5.00
3600.0 0.045
0.924

1
1
0
0
CS
3604.2
0.926 0.313
6.00
3608.5 0.038
0.927
3613.1
0.928 0.347
7.00
3618.0 0.031
0.929
3623.0
0.931 0.374
8.00
3628.3 0.022
0.932
3633.8
0.933 0.390
9.00
3639.5 0.011
0.935
10001 1001 10002 1002 10003 1003 10004 1004 10005 1005 10006 1006 10007 1007 10008 1008 10009 1009 10010

3645.4
0.396
2

0.937 10.00
0.000

My [kNm]
My [kNm]
My [kNm]
My [kNm]
σc,b [MP]
Result
3651.6
0.938
3657.8
0.940 0.390
11.00
Stress Calculation at a Rectangular Prestressed Concrete CS

3663.8 -0.011
0.941
3669.6
0.943 0.374
1.000 = 1368 N/mm2

12.00
3675.1 -0.022

Table 2: Results
0.944
3680.5 0.944
0.944 0.347
3682.2 13.00
3678.8 -0.031
0.942
3673.9
0.313
Figure 5: Tendon Geometry

0.941 14.00
3669.2 -0.038

−156.08
−4.51
−1406.08
−11.76
−185.88
−4.82
−1435.88
−12.47
SOF.
0.940
3664.8
0.939 0.271
15.00
3660.6 -0.045
0.938
3656.6

Figure 6: Prestress Forces and Stresses

0.937 0.223
16.00
3652.8 -0.050
0.936
3649.2
0.935 0.171
17.00
3645.9 -0.055
0.934
3642.9
0.934 0.115
18.00
3640.4 -0.057

−156.11
−4.51
−1406.11
−11.76
−185.91
−4.82
−1435.91
−12.47
Ref.
0.933
3638.2
0.933 0.057
19.00
3636.3 -0.057
0.932
10011 1010 10012 1011 10013 1012 10014 1013 10015 1014 10016 1015 10017 1016 10018 1017 10019 1018 10020

3634.3
0.000
3

20.00
-0.057
0.944
0.932

5
 Stress Calculation at a Rectangular Prestressed Concrete CS

4 Design Process1
Design with respect to DIN EN 1992-1-1:2004 (NA) [1] [2]:2

Material:

3.1: Concrete Concrete: C 35/ 45

3.1.2: Tab. 3.1: Ecm for C 35/ 45 Ecm = 34000 N/ mm2

3.3: Prestressing Steel Prestressing Steel: Y 1770

3.3.6 (3): Ep for wires Ep = 195000 N/ mm2

3.3.2, 3.3.3: ƒpk Characteristic tensile ƒpk = 1770 N/ mm2

strength of prestressing steel
ƒp0,1k = 1520 N/ mm2
3.3.2, 3.3.3: ƒp0,1k 0.1% proof-stress of
prestressing steel, yield strength
Prestressing system: BBV L19 150 mm2

19 wires with area of 150 mm2 each, giving a total of Ap = 28.5 cm2

Cross-section:

Ac = 1.0 · 1.0 = 1 m2

Diameter of duct ϕdct = 97 mm

Ratio αE,p = Ep / Ecm = 195000 / 34000 = 5.74

Ac,netto = Ac − π · (ϕdct / 2)2 = 0.9926 m2

Ade = Ac + Ap · αE,p = 1.013 m2

The force applied to a tendon, i.e. the force at the active end during
5.10.2.1 (1)P: Prestressing force during tensioning, should not exceed the following value
tensioning - Maximum stressing force
5.10.2.1 (1)P: Eq. 5.41: Pm maximum Pm = Ap · σp,m
stressing force
where σp,m = min 0.80ƒpk ; 0.90ƒp0,1k

(NDP) 5.10.2.1 (1)P: σp,m maximum
stress applied to the tendon
Pm = Ap · 0.80 · ƒpk = 28.5 · 10−4 · 0.80 · 1770 = 4035.6 kN

Pm = Ap · 0.90 · ƒp0,1k = 28.5 · 10−4 · 0.90 · 1520 = 3898.8 kN

→ Pm = 3898.8 kN and σp,m = 1368 N/ mm2

The value of the initial prestress force at time t = t0 applied to the

concrete immediately after tensioning and anchoring should not exceed
5.10.3 (2): Prestress force the following value

5.10.3 (2): Eq. 5.43: Pm0 initial pre- Pm0 () = Ap · σp,m0 ()
stress force at time t = t0
1 The tools used in the design process are based on steel stress-strain diagrams, as

defined in [1] 3.3.6: Fig. 3.10, which can be seen in Fig 3.

2 The sections mentioned in the margins refer to DIN EN 1992-1-1:2004 (German Na-

tional Annex) [1], [2], unless otherwise specified.

6 Benchmark No. 17 | SOFiSTiK 2023

Stress Calculation at a Rectangular Prestressed Concrete CS

where σp,m0 () = min 0.75ƒpk ; 0.85ƒp0,1k


(NDP) 5.10.3 (2): σp,m0 () stress in the
tendon immediately after tensioning or
Pm0 = Ap · 0.75 · ƒpk = 28.5 · 10−4 · 0.75 · 1770 = 3783.4 kN transfer

Pm0 = Ap · 0.85 · ƒp0,1k = 28.5 · 10−4 · 0.85 · 1520 = 3682.2 kN

→ Pm0 = 3682.2 kN and σp,m0 = 1292 N/ mm2

Further calculations for the distribution of prestress forces and stresses

along the beam are not in the scope of this Benchmark and will not be
described here. The complete diagram can be seen in Fig. 5, after the
consideration of losses at anchorage and due to friction, as calculated
by SOFiSTiK. There the values of σp,m = 1368 N/ mm2 and Pm0 =
3682.2 kN can be visualised.

Load Actions:

Self weight per length: γ = 25 kN/ m

→ g1 = γ · A = 25 · 1 = 25 kNm

Safety factors at ultimate limit state DIN EN 1990/NA [4]: (NDP) A.1.3.1 (4):
Tab. NA.A.1.2 (B): Partial factors for ac-
tions
Actions (unfavourable) Safety factor at final state (NDP) 2.4.2.2 (1): Partial factors for pre-
• permanent γG = 1.35 stress

• prestress γP = 1.00

Combination coefficients at serviceability limit state

g1 = 25 kNm: for rare, frequent and quasi-permanent combination

(for stresses)

At  = 10.0 m middle of the span:

Mg = g1 · L2 / 8 = 1250 kNm

Np = Pm0 ( = 10.0 m) = −3653.0 kN (from SOFiSTiK)

Calculation of stresses σc,b at  = 10.0 m middle of the span: The concrete stresses may be deter-
mined for each construction stage un-
Position of the tendon: z = 0, 3901 m der the total quasi-permanent combina-
tion σc {G + Pm0 + ψ2 · Q}
In this Benchmark no variable load Q is
• Case : Prestress at construction stage section 0 (P cs0)
examined

−σc

Mp
Np
zp
Pm0,=10

+σc

Np = −3653.0 kN

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 Stress Calculation at a Rectangular Prestressed Concrete CS

Mp1 = NP · z = −3653.0 · 0.3901 = −1425.04 kNm

zs the new position of the center of the
cross-section for cs0 Mp2 = NP · zs = −3653.0 · 0.002978 = −10.879 kNm
zp = z + zs

Mp bending moment caused by pre- Mp = −1425.04 − 10.879 = −1435.91 kNm = My

stressing
Np My
W1,cs0 cross-section moduli for con- σc,b = +
truction stage 0 at the bottom left and Ac,netto W1,cs0
right point
σc,b stress at the concrete at the bottom −3653.0 −1435.91
of the cross section
σc,b = + = −12.47 MP
0.9926 0.1633

• Case : Prestress and self-weight at con. stage sect. 0 (P+G cs0)

−σc

Mg + Mp
Np
zp
Pm0,=10

+σc

Np = −3653.0 kN and Mg = 1250 kNm

As computed above: Mp = −1435.91 kNm

My = 1250 − 1435.91 = −185.91 kNm

−3653.0 −185.91
σc,b = + = −4.82 MP
0.9926 0.1633

• Case : Prestress at con. stage sect. 1 (P cs1)

Np = −3653.0 kN and Mp1 = −1425.04 kNm (as above)

Mp2 = NP · zs = −3653.0 · (−0.00518) = 18.92 kNm

Mp = −1425.04 + 18.92 = −1406.11 kNm = My

Np My
W1,cs1 cross-section moduli for con- σc,b = +
truction stage 1 Ade W1,cs1

−3653.0 −1406.11
σc,b = + = −11.76 MP
1.013 0.172

• Case V: Prestress and self-weight at con. stage sect. 1 (P+G cs1)

Np = −3653.0 kN and Mg = 1250 kNm

As computed above: Mp = −1406.11 kNm

My = 1250 − 1406.11 = −156.11 kNm

−3653.0 −156.11
σc,b = + = −4.51 MP
1.013 0.172

8 Benchmark No. 17 | SOFiSTiK 2023

Stress Calculation at a Rectangular Prestressed Concrete CS

5 Conclusion
This example shows the calculation of the stresses, developed in the concrete cross-section due to
prestress and bending. It has been shown that the results are reproduced with excellent accuracy.

6 Literature
[1] DIN EN 1992-1-1/NA: Eurocode 2: Design of concrete structures, Part 1-1/NA: General rules and
rules for buildings - German version EN 1992-1-1:2005 (D), Nationaler Anhang Deutschland - Stand
Februar 2010. CEN. 2010.
[2] F. Fingerloos, J. Hegger, and K. Zilch. DIN EN 1992-1-1 Bemessung und Konstruktion von
Stahlbeton- und Spannbetontragwerken - Teil 1-1: Allgemeine Bemessungsregeln und Regeln für
den Hochbau. BVPI, DBV, ISB, VBI. Ernst & Sohn, Beuth, 2012.
[3] Beispiele zur Bemessung nach Eurocode 2 - Band 1: Hochbau. Ernst & Sohn. Deutschen Beton-
und Bautechnik-Verein E.V. 2011.
[4] DIN EN 1990/NA: Eurocode: Basis of structural design, Nationaler Anhang Deutschland DIN EN
1990/NA:2010-12. CEN. 2010.

SOFiSTiK 2023 | Benchmark No. 17 9